Question: Lara's radar on her submarine detects two other subs nearby. The first sub is $360 \text{ m}$ away, the second sub is $250 \text{ m}$, and the angle between them is $34^\circ$. What is the distance between the other two subs? Do not round during your calculations. Round your final answer to the nearest meter.
Explanation: Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $AB=d$. $\;\;34^\circ$ $d$ $360\text{ m}$ $250\text{ m}\,\,\,$ $A$ $B$ $C$ Since we are given two side lengths and the angle measure between them, we can use the law of cosines. Using the law of cosines $\begin{aligned} (AB)^2&=(AC)^2+(BC)^2-2AC\!\cdot\! BC\!\cdot\!\cos(C)\\\\ d^2&=250^2+360^2-2\cdot250\cdot360\cdot\cos(34^\circ) \gray{\text{Substitute}}\\\\ d&=\sqrt{250^2+360^2-2\cdot250\cdot360\cdot\cos(34^\circ)}\\\\ d&\approx 207 \end{aligned}$ The answer The distance between the other two subs is $207$ meters.